Reciprocal Class of Jump Processes

Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set $$\mathcal {A}\subset \mathbb {R}^{d}$$A⊂Rd. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of $$\mathcal {A}$$A plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.

[1]  Etienne Pardoux,et al.  Differential Calculus and Integration by Parts on Poisson Space , 1990 .

[2]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[3]  Christian L'eonard,et al.  Reciprocal processes. A measure-theoretical point of view , 2013, 1308.0576.

[4]  Bernard C. Levy,et al.  Characterization of Multivariate Stationary Gaussian Reciprocal Diffusions , 1997 .

[5]  Paolo Dai Pra,et al.  A stochastic control approach to reciprocal diffusion processes , 1991 .

[6]  Nicolas Privault,et al.  Markovian bridges and reversible diffusion processes with jumps , 2004 .

[7]  J. Zambrini,et al.  Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus , 1991 .

[8]  Arthur J. Krener,et al.  Reciprocal diffusions and stochastic differential equations of second order , 1988 .

[9]  J. Zambrini,et al.  Symmetries in the stochastic calculus of variations , 1997 .

[10]  Reciprocal Processes: A Stochastic Analysis Approach , 2014 .

[11]  Jesús A. De Loera,et al.  Algebraic and Geometric Ideas in the Theory of Discrete Optimization , 2012, MOS-SIAM Series on Optimization.

[12]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[13]  M. Thieullen.  Second order stochastic differential equations and non-Gaussian reciprocal diffusions , 1993 .

[14]  Bridges of Markov counting processes. Reciprocal classes and duality formulas , 2014, 1408.1332.

[15]  R. Murr.  Bridges of Markov Counting Processes , 2014 .

[16]  A characterization of reciprocal processes via an integration by parts formula on the path space , 2002 .

[17]  C. Stein.  Approximate computation of expectations , 1986 .

[18]  Edward Nelson.  Dynamical Theories of Brownian Motion , 1967 .

[19]  Louis H. Y. Chen.  Poisson Approximation for Dependent Trials , 1975 .

[20]  B. Jamison.  Reciprocal processes , 1974 .

[21]  J. Tyrrell,et al.  ALGEBRAIC NUMBER THEORY , 1969 .

[22]  B. Jamison,et al.  Reciprocal Processes: The Stationary Gaussian Case , 1970 .

[23]  A. Krener.  Reciprocal diffusions in flat space , 1997 .

[24]  A. Krener,et al.  Dynamics and kinematics of reciprocal diffusions , 1993 .

[25]  J. Zambrini.  Variational processes and stochastic versions of mechanics , 1986 .

[26]  Anton Wakolbinger,et al.  A simplified variational characterization of Schrödinger processes , 1989 .

[27]  S. Roelly,et al.  Duality formula for the bridges of a Brownian diffusion: Application to gradient drifts , 2005 .

[28]  B. Jamison.  The Markov processes of Schrödinger , 1975 .

[29]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[30]  J. Mecke,et al.  Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen , 1967 .

[31]  S. C. Chay,et al.  On quasi-Markov random fields , 1972 .